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Habilitation à diriger des recherches

Non linear models from relativistic quantum mechanics : spectral and asymptotic analysis and related problems.

Abstract : This thesis is devoted to some problems from relativistic and non relativistic mechanics.In a first part, I describe my work on spectral pollution. I present first the results on the stability by perturbation of this phenomenon from numerical spectral theory. Then I detail the analysis of two methods of approximation of the spectrum free from any pollution : the quadratic projective method applied to Dirac operators and the Davies-Plum method applied, among others, to the Maxwell operator in a bounded cavity.In a second part, I present two analysis on the dispersive properties of the Dirac operator. The first one is on Kato smoothness estimates for coulombic type perturbations obtained by Mourre's methods. The second one is on Morawetz estimates for magnetic perturbations.The third part describes the results on the bilinear control of Schr\"odinger equations. It is essentially results on approximate controllability with low time regularity and non controllability. Some quantitative results on the time and energy control are also presented.The last part describes the analysis of the stability of stationary solutions of non linear Dirac equations. An analysis of the spectral properties of the linearisation gives results on the linear stability while the analysis of non linear resonances gives asymptotic stability properties.
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Habilitation à diriger des recherches
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https://tel.archives-ouvertes.fr/tel-01094575
Contributor : Nabile Boussaid Connect in order to contact the contributor
Submitted on : Friday, December 12, 2014 - 3:26:19 PM
Last modification on : Monday, October 11, 2021 - 10:04:09 AM
Long-term archiving on: : Saturday, April 15, 2017 - 8:09:16 AM

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Nabile Boussaid. Non linear models from relativistic quantum mechanics : spectral and asymptotic analysis and related problems.. Analysis of PDEs [math.AP]. université de Franche-Comté, 2014. ⟨tel-01094575⟩

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